{
  "id": "1308.6812",
  "version": "v1",
  "published": "2013-08-30T17:58:52.000Z",
  "updated": "2013-08-30T17:58:52.000Z",
  "title": "A classification of nilpotent 3-BCI groups",
  "authors": [
    "Hiroki Koike",
    "István Kovács"
  ],
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "Given a finite group $G$ and a subset $S\\subseteq G,$ the bi-Cayley graph $\\bcay(G,S)$ is the graph whose vertex set is $G \\times \\{0,1\\}$ and edge set is $\\{\\{(x,0),(s x,1)\\} : x \\in G, s\\in S \\}$. A bi-Cayley graph $\\bcay(G,S)$ is called a BCI-graph if for any bi-Cayley graph $\\bcay(G,T),$ $\\bcay(G,S) \\cong \\bcay(G,T)$ implies that $T = g S^\\alpha$ for some $g \\in G$ and $\\alpha \\in \\aut(G)$. A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs.In this paper we prove that, a finite nilpotent group is a 3-BCI-group if and only if it is in the form $U \\times V,$ where $U$ is a homocyclic group of odd order, and $V$ is trivial or one of the groups $\\Z_{2^r},$ $\\Z_2^r$ and $\\Q_8$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-30T17:58:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25"
    ],
    "keywords": [
      "bi-cayley graph",
      "classification",
      "finite nilpotent group",
      "finite group",
      "homocyclic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.6812K"
    }
  }
}